Nonuniqueness of the $2$D Euler Evolutions with L1∩ Lp Vorticity
Abstract
For each prescribed 1<p<∞, we construct a compactly supported initial vorticity in L1(ℝ2)∩ Lp(ℝ2) that admits two distinct evolutions under the unforced two-dimensional incompressible Euler equation on [0,1]. Both vorticities are strongly continuous in L1∩ Lp and are transported exactly by their respective almost-everywhere invertible, measure-preserving regular Lagrangian flows. The construction couples unstable four-vortex configurations across increasingly small spatial scales, realizing them through concentrated signed vortex patches. At each finite cutoff, a prepared reference evolution remains close to rotating configurations, while a second evolution differs initially only at the finest scale. In the second evolution, the finest configuration consists of separating dipoles whose remote strain activates the instability of the next larger configuration. This process propagates through successive scales with summable transition times. As the cutoff tends to infinity, the two families of initial data converge strongly, along a subsequence, to the same datum, while a nonzero difference survives at a fixed larger scale and is detected by a fixed smooth observable at a positive limiting time. Quantitative concentration and relative comparison estimates establish this cascade for exact Euler patch solutions. Strong compactness through inverse-flow stability then preserves both exact vorticity transport and the positive-time distinction between the two limiting evolutions. The initial datum and construction parameters may depend on p.
Related
- Nonuniqueness of Solutions to 2D Incompressible Euler Equations with Unbounded Vorticity
- A Sharp Uniqueness-Nonuniqueness Dichotomy for Neutral Vortex Arrays in $2$D Euler Equations
- Flexibility of two-dimensional Euler flows with integrable vorticity
- Vorticity blowup in 2D compressible Euler equations
- An improved bound on the support diameter of nonnegative planar Euler vorticity
BibTeX
@misc{hou2026nonuniqueness,
title = {{Nonuniqueness of the $2$D Euler Evolutions with $L^1\cap L^p$ Vorticity}},
author = {Thomas Y. Hou and Xiang Qin and Peicong Song and Wenhan Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40553},
eprint = {2610.40553},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40553},
}
RIS
TY - UNPB
TI - Nonuniqueness of the $2$D Euler Evolutions with L^1∩ L^p Vorticity
AU - Thomas Y. Hou
AU - Xiang Qin
AU - Peicong Song
AU - Wenhan Zhang
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40553
UR - https://arxiv.org/abs/2610.40553
AB - For each prescribed 1<p<∞, we construct a compactly supported initial vorticity in L^1(ℝ^2)∩ L^p(ℝ^2) that admits two distinct evolutions under the unforced two-dimensional incompressible Euler equation on [0,1]. Both vorticities are strongly continuous in L^1∩ L^p and are transported exactly by their respective almost-everywhere invertible, measure-preserving regular Lagrangian flows. The construction couples unstable four-vortex configurations across increasingly small spatial scales, realizing them through concentrated signed vortex patches. At each finite cutoff, a prepared reference evolution remains close to rotating configurations, while a second evolution differs initially only at the finest scale. In the second evolution, the finest configuration consists of separating dipoles whose remote strain activates the instability of the next larger configuration. This process propagates through successive scales with summable transition times. As the cutoff tends to infinity, the two families of initial data converge strongly, along a subsequence, to the same datum, while a nonzero difference survives at a fixed larger scale and is detected by a fixed smooth observable at a positive limiting time. Quantitative concentration and relative comparison estimates establish this cascade for exact Euler patch solutions. Strong compactness through inverse-flow stability then preserves both exact vorticity transport and the positive-time distinction between the two limiting evolutions. The initial datum and construction parameters may depend on p.
ER -